6.2 Percentages, Ratio, Proportion & Averages

Percentages: Of, Increase and Decrease

Percentage items dominate the Numerical Ability subtest. Percent means 'per hundred', so 35% = 35/100 = 0.35. Three operations cover almost every question: finding a percent of a number, computing a percent increase or decrease, and finding what percent one number is of another.

Percent of a number

'Percent of' translates directly to multiplication: part = percent x whole. Convert the percent to a decimal or an easy fraction first.

Worked example: 35% of 240. Method 1 (decimal): 0.35 x 240 = 84. Method 2 (mental, no calculator): 10% of 240 = 24, so 30% = 72; 5% is half of 10% = 12; 72 + 12 = 84. Splitting a percent into 10% and 5% chunks is the fastest no-calculator technique on this exam.

Handy anchors: 10% moves the decimal one place left; 1% moves it two places left; 50% is half; 25% is a quarter; 20% is one-fifth. Build any percent from these.

Percent increase and decrease

A decrease of r% leaves (100 - r)% of the original; an increase of r% gives (100 + r)%. Rather than compute the change and risk forgetting to add or subtract it, use the shortcut multiplier.

Worked example: A shirt priced PHP 800 at 25% off. Discount = 25% of 800 = 200, so sale price = 800 - 200 = 600. Shortcut: 25% off means paying 75%, and 0.75 x 800 = 600 in one step.

Worked example (increase): A PHP 1,200 salary rises 15%. New salary = 1.15 x 1,200 = 1,380.

Successive-percentage trap: two discounts of 20% and 10% are NOT a 30% discount. Apply them one at a time: after 20% off, PHP 1,000 becomes 800; then 10% off 800 is 720, an effective 28% discount. Percentages of different bases never simply add.

Percent one number is of another / percent change

To find what percent A is of B, compute (A / B) x 100. For percent profit or loss the base is almost always the cost: percent profit = (profit / cost) x 100.

Worked example: A vendor buys at PHP 250 and sells at PHP 325. Profit = 325 - 250 = 75. Percent profit = (75 / 250) x 100 = 0.30 x 100 = 30%. A frequent trap is dividing by the selling price (325) instead of the cost. Read carefully to see whether the base is cost or selling price.

Ratio and proportion

A ratio compares quantities: 3:5 is read '3 to 5', and the parts total 3 + 5 = 8 shares. A proportion states that two ratios are equal and is solved by cross-multiplication.

Worked example: The ratio of male to female employees is 3:5 with 40 employees total. Total shares = 8, so one share = 40 / 8 = 5. Female = 5 shares x 5 = 25; male = 3 x 5 = 15. Check: 25 + 15 = 40.

Worked example (proportion): If 4 notebooks cost PHP 100, how much do 10 cost? Set 4/100 = 10/x, so 4x = 1,000 and x = 250. When setting up a proportion, keep the same units in matching positions (notebooks over pesos on both sides), or the cross-multiplication gives a wrong answer.

Direct and inverse variation

In direct variation two quantities rise and fall together (y = kx): more items, more cost. In inverse variation one rises as the other falls (y = k/x): more workers, fewer days.

Worked example (inverse): If 5 workers finish a job in 12 days, how many days will 10 workers take? Total work = 5 x 12 = 60 worker-days. With 10 workers: 60 / 10 = 6 days. Doubling the workforce halves the time, the hallmark of inverse variation. The trap answer doubles to 24 days by wrongly treating it as direct variation.

Averages: mean, median, mode

The mean (average) is the sum divided by the count. The median is the middle value when the data are ordered (average the two middle values if the count is even). The mode is the most frequent value.

Worked example (missing score): A student scored 80, 75, and 90 and wants an average of 85 over four tests. Required total = 85 x 4 = 340. Current total = 80 + 75 + 90 = 245. Fourth score = 340 - 245 = 95. The key move is turning the target average into a required total (average x count), then subtracting what is already there.

Worked example (why you cannot average averages): if classes of 20 and 30 students average 80 and 90, the combined average is (20 x 80 + 30 x 90) / 50 = (1,600 + 2,700) / 50 = 4,300 / 50 = 86, not 85. Never simply average the two averages when the groups differ in size.

Worked example (percent change between two values): sales rise from PHP 400 to PHP 500. Percent increase = (change / original) x 100 = (100 / 400) x 100 = 25%. The base is the ORIGINAL value (400), not the new one; dividing by 500 wrongly yields 20% and is the most common distractor. When a value falls, the same formula gives the percent decrease.